 
 
 
13.5.2  Series expansion
The taylor or
series command finds Taylor expansions.
- 
taylor takes one mandatory and
four optional arguments:
- 
expr, an expression depending on a variable.
- Optionally, x, the variable (by default x).
- Optionally n, an integer, the order of the series expansion
(by default 5).
- Optionally, a, the center of the Taylor expansion (by
default 0). This can be combined with the optional x by
replacing x by x=a.
- dir, a direction, which can be -1 or
1, for unidirectional series expansion, or 0 (for
bidirectional series expansion) (by default 0).
 
- taylor(expr,x ⟨,a,n,dir ⟩)
returns the Taylor expansion of expr about a or order n;
consisting of a polynomial in x−a plus a remainder of the form
of the form:
where order_size is a function such that
| ∀ r>0, |  | xr order_size(x)=0 |  
 
For regular series expansion, order_size
is a bounded function, but for non regular series expansion, it might tend slowly to
infinity, for example like a power of ln(x).
Example
or:
or (be careful with the order of the arguments):
or:
|  | | | sin | ⎛ ⎝
 | 1 | ⎞ ⎠
 | +cos | ⎛ ⎝
 | 1 | ⎞ ⎠
 | ⎛ ⎝
 | x−1 | ⎞ ⎠
 | − |  | sin | ⎛ ⎝
 | 1 | ⎞ ⎠
 | ⎛ ⎝
 | x−1 | ⎞ ⎠
 | 2+ | ⎛ ⎝
 | x−1 | ⎞ ⎠
 | 3 order_size | ⎛ ⎝
 | x−1 | ⎞ ⎠
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
Remark.
The order returned by taylor may
be smaller than n if cancellations between numerator and denominator
occur, for example consider
| taylor(x^3+sin(x)^3/(x-sin(x)),x=0,5) | 
|  | | | 6− |  | x2+x3+ |  | x4+x6 order_size | ⎛ ⎝
 | x | ⎞ ⎠
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
which is only a 2nd degree expansion.
Indeed the numerator and denominator valuation is 3, hence you lose 3
orders. To get order 4, you should use n=7.
| taylor(x^3+sin(x)^3/(x-sin(x)),x=0,7) | 
|  | | | 6− |  | x2+x3+ |  | x4− |  | x6+x8 order_size | ⎛ ⎝
 | x | ⎞ ⎠
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
a fourth degree expansion.
Examples
Find a 4th-order expansion of cos(2x)2 in the vicinity of
x=π/6.
| taylor(cos(2*x)^2,x=pi/6, 4) | 
|  | | |  |  | − | √ |  |  | ⎛ ⎜
 ⎜
 ⎝
 | x− |  | ⎞ ⎟
 ⎟
 ⎠
 | +2 | ⎛ ⎜
 ⎜
 ⎝
 | x− |  | ⎞ ⎟
 ⎟
 ⎠
 |  | + |  |  | √ |  |  | ⎛ ⎜
 ⎜
 ⎝
 | x− |  | ⎞ ⎟
 ⎟
 ⎠
 |  | − |  |  | ⎛ ⎜
 ⎜
 ⎝
 | x− |  | ⎞ ⎟
 ⎟
 ⎠
 |  | + | ⎛ ⎜
 ⎜
 ⎝
 | x− |  | ⎞ ⎟
 ⎟
 ⎠
 |  | order_size | ⎛ ⎜
 ⎜
 ⎝
 | x− |  | ⎞ ⎟
 ⎟
 ⎠
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
Find a 5th-order series expansion of arctan(x) in the vicinity of
x=+∞.
| series(atan(x),x=+infinity,5) | 
|  | | |  |  | − |  | + |  | − |  | + | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 |  | order_size | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
Note that the expansion variable and the argument of the
order_size function is
 h=1/x → 0  as x→+∞.
Find a 2nd-order expansion of (2x−1)e1/x−1 in the vicinity of
x=+∞.
| series((2*x-1)*exp(1/(x-1)),x=+infinity,3) | 
Output (only a 1st-order series expansion):
|  | | | 2 | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 |  | +1+ |  | + |  |  | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 |  | + | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 |  | order_size | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
Note that this is only a 1st-order expansion. To get a 2nd-order
series expansion in 1/x:
| series((2*x-1)*exp(1/(x-1)),x=+infinity,4) | 
|  | | | 2 | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 |  | +1+ |  | + |  |  | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 |  | + |  |  | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 |  | + | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 |  | order_size | ⎛ ⎜
 ⎜
 ⎝
 |  | ⎞ ⎟
 ⎟
 ⎠
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
Find a 2nd-order series expansion of (2x−1)e1/x−1 in the vicinity
of x=-∞.
| series((2*x-1)*exp(1/(x-1)),x=-infinity,4) | 
|  | | | −2 | ⎛ ⎜
 ⎜
 ⎝
 | − |  | ⎞ ⎟
 ⎟
 ⎠
 |  | +1+ |  | + |  |  | ⎛ ⎜
 ⎜
 ⎝
 | − |  | ⎞ ⎟
 ⎟
 ⎠
 |  | − |  |  | ⎛ ⎜
 ⎜
 ⎝
 | − |  | ⎞ ⎟
 ⎟
 ⎠
 |  | + | ⎛ ⎜
 ⎜
 ⎝
 | − |  | ⎞ ⎟
 ⎟
 ⎠
 |  | order_size | ⎛ ⎜
 ⎜
 ⎝
 | − |  | ⎞ ⎟
 ⎟
 ⎠
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
Find a 2nd-order series expansion of (1+x)1/x/x3  in
the vicinity of x=0+.
| series((1+x)^(1/x)/x^3,x=0,2,1) | 
(Note that this is a one-sided series expansion, since dir=1.)
|  | | | e x−3− |  | x−2+x−1 order_size | ⎛ ⎝
 | x | ⎞ ⎠
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
 
 
